Find the center and foci of the ellipse:

[tex]\[
9x^2 + 16y^2 + 126x + 96y + 441 = 0
\][/tex]

Center: [tex]\([ \, ? \, , \, ? \, ]\)[/tex]

Foci: [tex]\([ \, \square \pm \sqrt{ \, ? \, } \, , \, ? \, ]\)[/tex]



Answer :

To find the center and foci of the given ellipse, we need to rewrite the given equation in its standard form, which typically looks like:

[tex]\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \][/tex]

where [tex]\((h, k)\)[/tex] is the center of the ellipse, and [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the semi-major and semi-minor axes, respectively.

Given the equation:

[tex]\[ 9x^2 + 16y^2 + 126x + 96y + 441 = 0 \][/tex]

We need to complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.

### Completing the Square

First, we group the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:

[tex]\[ 9x^2 + 126x + 16y^2 + 96y + 441 = 0 \][/tex]

#### For [tex]\(x\)[/tex]-terms:
[tex]\[ 9(x^2 + \frac{126}{9}x) = 9(x^2 + 14x) \][/tex]

To complete the square for [tex]\(x\)[/tex]:

[tex]\[ x^2 + 14x \][/tex]
[tex]\[ = (x + 7)^2 - 49 \][/tex]

So,

[tex]\[ 9(x + 7)^2 - 9 \cdot 49 \][/tex]

#### For [tex]\(y\)[/tex]-terms:
[tex]\[ 16(y^2 + \frac{96}{16}y) = 16(y^2 + 6y) \][/tex]

To complete the square for [tex]\(y\)[/tex]:

[tex]\[ y^2 + 6y \][/tex]
[tex]\[ = (y + 3)^2 - 9 \][/tex]

So,

[tex]\[ 16(y + 3)^2 - 16 \cdot 9 \][/tex]

### Substitute Back:

[tex]\[ 9(x + 7)^2 - 9 \cdot 49 + 16(y + 3)^2 - 16 \cdot 9 + 441 = 0 \][/tex]

Simplify the constants:

[tex]\[ 9(x + 7)^2 - 441 + 16(y + 3)^2 - 144 + 441 = 0 \][/tex]

Combine the constants:

[tex]\[ 9(x + 7)^2 + 16(y + 3)^2 - 144 = 0 \][/tex]

### Standard Form:

[tex]\[ 9(x + 7)^2 + 16(y + 3)^2 = 144 \][/tex]

Divide through by 144:

[tex]\[ \frac{(x + 7)^2}{16} + \frac{(y + 3)^2}{9} = 1 \][/tex]

Now we have the standard form of the ellipse equation:

[tex]\[ \frac{(x + 7)^2}{16} + \frac{(y + 3)^2}{9} = 1 \][/tex]

From this, we can identify:

- The center [tex]\((h, k)\)[/tex] of the ellipse is [tex]\((-7, -3)\)[/tex].
- The lengths of the semi-major axis [tex]\(a\)[/tex] and the semi-minor axis [tex]\(b\)[/tex]:
- [tex]\(a^2 = 16\)[/tex] so [tex]\(a = 4\)[/tex]
- [tex]\(b^2 = 9\)[/tex] so [tex]\(b = 3\)[/tex]

### Determining the Foci:

The distance [tex]\(c\)[/tex] from the center to each focus is found using:

[tex]\[ c = \sqrt{a^2 - b^2} \][/tex]

Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

[tex]\[ c = \sqrt{16 - 9} = \sqrt{7} \][/tex]

### Foci Coordinates:

The foci are along the [tex]\(x\)[/tex]-axis (horizontal) because [tex]\(a > b\)[/tex], so we add and subtract [tex]\(c\)[/tex] from the [tex]\(x\)[/tex]-coordinate of the center:

The foci are at:

[tex]\[ (-7 + \sqrt{7}, -3) \text{ and } (-7 - \sqrt{7}, -3) \][/tex]

### Answers:

- Center: [tex]\((-7, -3)\)[/tex]
- Foci: [tex]\((-7 \pm \sqrt{7}, -3)\)[/tex]